Servo DC Motor Position Control Based on Sliding Mode Approach

Ahmed. M. Kassem* and Ali M. Yousef**

* Control Technology Dep., Bini-Swief University, Egypt ** Electrical Dept., Assiut University, Egypt

Abstrac - A novel controller based on the sliding mode models the desired closed loop performance. Secondly, (SM) approach is designed for controlling DC motor in a control law is designed such that the system state a servo drive. The modeling and analysis of the servo trajectory is forced toward the sliding surface. The DC motor are obtained. The sliding mode controller system state trajectory in the period of time before (SMC) design changes such that its performance is reaching the sliding surface is called the reaching substantially improved. To improve the controller phase. The system dynamics in the reaching phase is performance in steady stat (zero error) the integral still influenced by uncertainties. Ideally, the switching sliding mode controller (ISMC) is used. Since the main of the control should occur at infinitely high frequency drawback of SMC is a phenomenon, the so-called to eliminate the deviation from sliding manifold. In chattering and resulting from discontinuous practice, the switching frequency is not infinitely high controllers. An ISMC with switched gains is used for due to the finite switching time. Thus, undesirable chattering reduction and controller robustness. For chattering appears in the control effort. Chattering is comparison, the proposed ISM with switched gains is highly undesirable because it excites unmodeled high compared with that of a PID controller. Digital frequency plant dynamics and this can cause simulations have been carried out in order to validate unforeseen instability [9]. Different studies tried to the effectiveness of the proposed scheme. The proposed solve this problem by combining fuzzy or neural controller offers very good tracking; it is highly robust, controller with the sliding mode [10,11]. reaches the final position very fast. Furthermore the application of the SM ensures reduction of the system In this paper, the position control of the DC motor order by one. Also, quick recovery from matched in servo drive has been developed based on SM disturbance in addition to good tracking ability is approach. A novel scheme using Integral sliding mode evaluated. Moreover, this scheme is robust against the (ISM) controller with switched gains has been parameters variations and eliminates the influence of investigated. The system responses are compared when modeling. PID controller is applied to the system. A PID controller is selected because the cost of Keywords : Servo DC motor, integral sliding mode implementation is inexpensive and is widely used in control, PID controller. industry. SMC methods yield nonlinear controllers which are robust against unmodeled dynamics and, 1. Introduction internal and external perturbations. Computer AC and DC servomotors [1-3] are in use in many simulations are performed to show the validity of the applications. Particularly, DC ones are used in proposed system. The results obtained with ISMC with computer peripherals and robot manipulators and are changed gains are compared with the traditional SMC characterized by: ability to produce full continuous and PID controller. The advantages and limitations of torque, controlled braking is relatively simple and low each method are discussed. cost as compared with similar AC drives at high powers. Sliding mode was introduced 2. Plant Description for the first time in the context of the variable structure systems (VSS). It has become so popular that now it The plant consists of a DC motor with an inertial represents this class of control systems. Even though, load. The DC motor is separately excited and in its early stages of development, the SMC theory was controlled, which schematic diagram is shown in Fig.1. overlooked because of the development in the famous In this section we proposed to design a controller to linear control theory. Recently, the variable structure control the motor-load angle speed. The system control strategy using the sliding-mode has been parameters are shown in appendix. A block diagram of focused on many studies and research for the control of the DC servomotor with automatic voltage regulator the DC servo drive systems [4-8]. The sliding-mode (AVR) is shown in Fig.2. control can offer many good properties, such as good performance against unmodeled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response [9]. These advantages of the sliding-mode control may be employed in the position and speed control of an DC servo system.

The design of SMC consists of two main steps. Firstly; one can select a sliding surface that Fig.1: DC Servomotor circuit diagram ˺

The state equations that describe the DC servomotor C a positive constant behavior [12] are: From the second theorem of Lyapunov, the stability condition can be written as: 2   2 d  K m  K f d  1  d       ia TL (1)  K (8) dt 2 J J dt 2 dt  dia   Ra  K b d  Va 1  2  ia (2) Where: 0 (positive definite) is a Lyapunov dt La La dt La 2 function and K a positive constant. The control voltage Let   command is calculated by substituting and  in  V u (3) equation (8) as follows [13]: a

   2   Substituting (1) into (2) and using (3) one gets:      BRa Km     r C r  C   2   2      R J  R J  K  d  BR K d  1 u a m  a m  (9) u m  a m    T (4) 2     L Km  1     Ra J m dt  Ra J m  dt J m  T K sign( )   J L  In state space matrix form, one gets: m A block diagram of this conventional SM controller is dx    A x B u D v (5) shown in Fig.3. dt Where   0 1  0     2     A  BR K , B K m , 0 a m        Ra J m Ra J m  0    1  D   (6)   J m    t    d    with x x x , x , x , v T Fig. 3: Block diagram of the conventional SM 1 2 1 2 dt L controller and The problem with this conventional controller is x1 position, that it has large chattering in the control output and the x rotor speed 2 drive is very noisy. Furthermore, because of chattering, u control input it is difficult to achieve small enough positioning error v disturbance in steady state [13]. To reduce chattering the sign t transpose function (infinite gain) of the conventional SM is substituted with a finite gain K within a small boundary. This affects controller's robustness too, but still the controller will remain robust enough if gain K is chosen large enough. An infinite or very large gain K increases chattering because the bandwidth of the automatic voltage regulator (AVR) is not infinite. We assumed an ideal AVR and did not include it in the plant model, but this assumption holds as long as the

bandwidth of the outer control loops does not exceed Fig. 2: Block diagram of the DC motor with AVR. that of AVR. The selection of a finite K gain affects the sliding surface. By choosing a finite, appropriate value for K the chattering is greatly reduced [13]. The

constant C basically determines the speed of response. 3. The Integral Sliding Mode (ISM) Controller It is the only parameter, which determines the dynamic

of the system in SM. The choice of the constant C is In this section we will show first the design of also bounded from the AVR bandwidth and the the SM controller and later, based on the analysis and encoder noise. Large values of constant C tend to faster interpretation of the controller block diagram we response, but if the bandwidth of the AVR is exceeded introduce the proposed ISM controller.  this will lead in chattering and it will deteriorate the The sliding surface is defined as:           controller performance. e C e r C r (7) Where To reduce greatly the steady state error an integral  block is added, which forces the system in steady state r the position reference

˻

toward a zero error positioning. This integral block   1  K(s) K (1 T s) (10) tends to slow down the transient response of the p T s D control system. For this reason it is switched on only I when the system approaches the final position. On one K P where K p , and K P TD represent the hand the sliding mode tracks the reference very fast TI ensuring a very small tracking error till the final proportional, integral and derivative gains of the positioning is reached and on the other hand the controller respectively. Define integral block, taking advantage of the small initial   1   1 T error, reduces it to zero very fast. The gain K is and I as the controller's n 2 T switched to a larger value K1 right before the position TI TD D command reaches the desired final value. The natural frequency and the damping coefficient, condition of switching is detected when speed and respectively. Then the PID transfer function can be acceleration signals of the position command have written as: opposite signs. This gain is returned to its previous  2    2 value immediately after the final position is reached.  n 2 n s s K(s) K P  (11) This ensures a fast and precise stop of the servo. If a 2 n s large gain will be applied all the time, chattering in the torque command will be inevitable. So, we have 5. Results and discussion designed a controller, which has a very good tracking and reaches very fast the final position. The controller The purpose of this part showing the validity and is also very robust to the outside disturbances or robustness of the proposed SMC approaches as applied parameter uncertainties. A block diagram of the ISMC to a DC servo system. Two SMC schemes are applied with switched gains is shown in Fig. 4. to control the position of the DC servomotor. Also, digital simulation and experimental setup are using to evaluate the model.

5.1. Digital Simulation Results Figure 5 shows the position and speed time responses for step change in the desired position angle  ( r) using SMC and PID controllers. It is seen that, applying a PID controller which its parameters has been tuned using the optimization method to the system when, there is no disturbance, the output completely tracks the desired position, but with the traditional SMC, the position achieved is slightly lower Fig. 4: Block diagram of the proposed ISM than the desired value. However, the settling time for controller with switched gains SMC is less (0.12 sec.) than the PID (0.47 sec.). To remove this obstacle the ISMC is applied to the 4. PID Controllers system. Fig. 6 shows the position and speed time responses using a PID controller and the classical SMC PID controllers are dominant and popular and, with matched disturbance, which shown in figure (6a). have been widely used because one can obtain the SMC is insensitive against the (matched) disturbance desired system responses and it can control a wide and the desired position is obtained even in the class of systems. This may lead to the thought that the presence of disturbance. A well-tuned PID controller PID controllers give solutions to all requirements, but may reduce the effect of the disturbance on the system. unfortunately, this is not always true [14]. Alternative However, by applying PID controller, it is impossible tuning methods have been recently presented including to reject the oscillations completely. The simulation disturbance reduction, magnitude optimum [15,16], results for ISMC when there is a random disturbance in pole placement and optimization methods [16,17]. the system is shown in Fig.7. The result shows the In this work, the PID optimal tuning method used exact position is obtained (without steady state error) ’t is found in [17]. In this method, the parameters of PID when ISMC is applied, and the disturbance doesn controller satisfying the constraints correspond to a affect the position. But when a PID controller is given domain in a plane. The optimal controller lies on applied, the position and speed of the servo drive does the curve. The design plot enables identification of the not reach to steady state but make oscillation around PID controller for desired robust conditions, and in the desired position. Appropriate PID can reduce this particular, gives the PID controller for lowest oscillation but it is impossible to reject the disturbance sensitivity. By applying this method, trade-off among effect completely. high frequency sensor noise, low frequency sensitivity, Since our concerns are also in robust stability against gain and phase margin constraints are also directly various model uncertainties, some system parameters available. have been changed in the following ways: The transfer function of a PID controller is given by:

˼

1. The moment of inertia is decreased by 10% to be 1.2e-5 Kg m2. 2. the armature rotor resistance, is increased by 20% to be 0.38 ohm. 3. The armature rotor inductance is increased by 15% to be 9.2e-5 H. For perturbed system the responses are shown in Fig.8. It should be seen that the system is robustly stable in spite of parameters variations. Moreover, the settling time and the overshoots calculation for position (c) angle and rotor speed is depicted in table 1. Fig. 6: Time response of the proposed ISMC driven DC servo motor with parameters changes

(a) (a)

(b) Fig. 5: Time response of the SMC and optimization

tuned PIDC driven DC servo motor. (b) Fig. 7: Time response of the proposed ISMC and PIDC driven DC servo motor with disturbance

(a) (a)

(b) (b)

˽

Fig. 8: Time response of the proposed ISMC and [7] M. F. Rahman, L. Zhong, M. Haque, and M. A. “A direct torque controlled Interior PIDC driven DC servo motor with Rahman, disturbance and parameters changes permanent synchronous ”, IEEE Trans. On Energy without a speed sensor Table 1: Settling time and overshoots calculation based Conversion, Vol. 18, No.1, March 2003, pp.17-22. “ A novel on control method techniques. [8] Jawad Faiz and S.Hossien Mohseni . Case Parameters PID SM ISM technique for estimation and control of flux control control control of a salient pole PMSM in DTC method based on ” IEEE Trans. On Indust. Appl., Vol. 50, Position Settling >>1 0.3 Sec. 0.2 Sec. MTPF. angle time (Sec.) Sec. +S.S.E No. 2, April 2003, pp. 262-270 Overshoots 3 rad. 2.9 rad. 3 rad. ” (rad.) [9] M. F. Rahman, E. Haque, and L. Zhong, Problems associated with the Rotor Settling >>1 0.23 Sec. 0.2 Sec. of an interior permanent magnet synchronous ”, IEEE Trans. On speed time (Sec.) Sec. motor drive and their remedies Overshoots 22 43 42 Indust. Electronics., Vol. 51, No. 4, August 2004, (rad./Sec.) rad/Sec. rad./Sec. rad./Sec. pp. 799-908. [10] Luis Romeral, Antoni Aris, Emiliano Aldabs ” Novel direct torque control 6. Conclusions and Marcel. G.Jayne. scheme with fuzzy adaptive torque ripple ” IEEE Trans. O In this paper, SMC, ISMC with switched gains and reduction. n Indust. Electronics, PID controllers have been considered for controlling Vol. 50, No. 3, June 2003, pp. 487-492. ” Direct torque the position of DC motor in servo drive. A comparison [11] N. R. Idris, and A. M. Yatim, method has been studied to show the relative control of induction with constant ”, advantages and limitations of each method. PID switching frequency and reduced torque ripple controllers are suitable if there is no disturbance in the IEEE Trans. On Indust.Electronics., Vol. 51, No. system. However, the settling time is longer than when 4, August 2004, pp. 758-767. SMC is applied to the system. Using an ISMC the [12] A. M. Abdel Ghany and Ahmed Bensenouci, desired position is obtained, whilst when the SMC is "Improved Free-chattering variable-structure for a used, the desired position is tracked only if one DC servomotor position control". 3rd Saudi considers an SMC with very high gain. Moreover ’t affect the system in the sliding Technical Conf., V 2, December 2004, pp. 21-30. disturbances don [13] Orges Gjini, Takaynki Kaneko and Hirosh mode. A novel scheme using Integral sliding mode Ohsawa. "A new controller for PMSM servo drive (ISM) controller with switched gains has been based on the sliding mode approach with investigated and evaluated in terms of less settling time parameter adaptation". IEE Trans. On IA, V 123, with no steady state error. No. 6, 2003, pp. 675-680. 7. References [14] Datta A., Ming-Tzu H. and Bhttacharyy S. P., "Structure and synthesis of PID controllers". [1] Ogata K., "Modern control Engineering". New Advanced in Industrial Control, Springer-Verlag Jersey, Prentice-Hall, 1990. London Limited, 2000. [2] EL Sharkawi M and Huang C., "Variable structure [15] Astrom K. J. and Hagglund T.., "PID controllers: tracking of DC motor for high performance theory, design and tuning". Instrument Society of applications". IEEE Trans. on Energy Conversion, America, 2nd edition, 1995. V 4, 1989, pp. 643-650. [16] Yaniv, O. and Nagurka, M., "Robust, PI controller [3] Ghazy M., "State of the art control techniques design satisfying sensitivity and uncertainty used in servo DC motor applications". Report: specifications", IEEE Trans. Automat. Control, V Elect. Dep., Helwan University, Egypt, 2002 . 48, 2003, pp. 2069-2072,. ” Influence of [4] M.E. Haque, and M.F. Rahman, [17] Kristiansson, B. and Lennartson, B., "Robust and stator resistance variation on controlled interior optimal tuning of PID controllers", IEE Proc. permanent magnet drive ”, IEEE Trans. Control theory and application, V 149, 2002, pp. performance and its compensation 17-25. On Energy Conversion , 2001, pp. 2563-2569. Appendix [5] Ciro Attaianese, Vito Nardi, Aldo Perfetto and “ Vectorial torque control: A Table 2. Parameter of the DC motor Giuseppe Tomasso. Parameters Values  novel approach to torque and flux control of Ra 3 ”, IEEE Trans. On Indust. Appl, µ La 150 H Vol. 35, No. 6, December 1999, pp. 1399-1405. ” Direct torque Km 6.1e-06 NV/A [6] Miran Rodic, Larel Jezernic , – Kv 6.102e-03V-s/rad control of PWM inverter fed ac motors a ”, IEEE Trans. On Indust. Electronics., Vol. Va 24 VDC survey 2 51, No. 4, August 2004, pp. 744-757. Jm 2.5e-5 Kg-m/s

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